Wave packets in the anomalous Ostrovsky equation

The anomalous Ostrovsky equation, which describes waves in vertically sheared ocean flows and magnetoacoustic waves, possesses steadily propagating, finite-amplitude, localized wave-packet solutions. It is shown here that these solutions can be obtained asymptotically, using Whitham modulation theory, as the solution to a nonlinear eigenvalue problem. This allows the various wave-packet solutions to be delineated and compared to solutions of the full equations of motion. A periodic solution with an embedded wave train is also constructed.

Figure 1

Typical solution of system (29) here for $c=-0.05$ showing the variables $r_2$ and $r_3$ coalescing at $X_{\mathrm{e}}$.

Figure 2

Values of (a) the wave-packet halfwidth $X_{\mathrm{e}}$ and (b) the edge phase ${\mathrm{\Theta }}_{\mathrm{e}}$, for speeds $-0.2\le c\le 1/3$.

Figure 3

Wave packets with (a) $N=7$ for $\epsilon =0.01954$, (b) $N=4$ for $\epsilon =0.03742$, and (c) $N=2$ for $\epsilon =0.09402$. The thin curve (red online) gives the WMT solution [with speeds (a) $c=-0.032$, (b) $c=-0.059$, and (c) $c=-0.134]$. The thicker curves give the upper and lower envelopes (blue online), and the full solution [with speeds (a) $c=-0.034$, (b) $c=-0.066$, and (c) $c=-0.166]$. The outer WMT solutions are terminated at $X=\pm X_0$, with (a) $X_0=\pm 2.3534$, (b) $X_0=\pm 2.4753$, and (c) $X_0=\pm 2.8031$ marked by solid dots.

Figure 4

Wave-packet speeds, $c$, as a function of the rotation strength, $\epsilon$, for various mode numbers, $N$. The dashed line gives the lower bound from (27). The dash-dotted curves give the speeds derived from Fig. 2 and (15) and the solid curves give the full solutions of (25). The dotted curve gives the single peak speed for $N=1$. The thicker solid curve in (b) (red online) gives the speed for large $\epsilon$ from the NLS approximation.

Figure 5

Wave packet from the continuation of the $N=1$ solution. Here $\epsilon =0.8$ and $c=-1.57306$. The NLS approximation is given by the thin curve (red online) and the upper and lower NLS envelopes are also shown. The thicker curve gives the full solution.

Figure 6

Periodic wave train containing $N=3$ wave packets. Here $\epsilon =0.05369$, $c=-0.09414$ and the period is $2X_0=5.2664$. The thin curve (red online) is the WMT solution and the thicker curve is the full solution.